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By RULE 2. The least common multiple of all the denominators is 12ax; therefore the fractions reduced to their least common denominator are

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4. Reduce to their least common denominator

7a 3x

5bx(a2-b2) 3a2bx-3ab2x 4a2b2+4ab3 3a3x-3ab2x ab(a3-b3)' ab(a-b3)' ab(a-3) ab(ab)

,

2. Reduce to a common denominator

(a-b) x (a+b)2x ab(ab2) x(a2—b2)' x(a2—b2)3 x(a2—b2)'

5x 3x 46

and

a-ba+b

ab

and

a+b'a-b'

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5. Reduce to their least common denominator

الاه

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ADDITION OF ALGEBRAIC FRACTIONS.

50. RULE. Reduce the fractions to a common denominator; then add their numerators, and under the sum write their common denominator; the fraction so formed will be the fraction required.

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all the fractions is a2-b; hence the fractions become

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SUBTRACTION OF ALGEBRAIC FRACTIONS.

51. RULE. Reduce the fractions to a common denominator, then take the difference of the numerators, under which write the common denominator, and the result will be the fraction required.

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MULTIPLICATION OF ALGEBRAIC FRACTIONS. 52. RULE. Multiply the numerators together for the numerator of the product, and the denominators together for the denominator of the product.

За

b

a-b

EXAMPLE. Multiply a+b by Here we have 3a xb =3ab for the numerator, and (a+b)×(a—b)=a2—b2 for the denominator; hence the product required is a2-62. NOTE. Cancel factors which appear in both terms.

3ab

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DIVISION OF ALGEBRAIC FRACTIONS. 53. RULE. Multiply the dividend by the divisor inverted, and the product will be the quotient required.

NOTE. The reason of the above rule will appear from the following example:—Let it be proposed to divide by: for write n,

n

then it will be where the value will not be altered by multiplying

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nd

numerator and denominator by d, which gives -=nx; but n is

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с

X which is the rule.

equal to 7, therefore nx 2:

d a

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7. What is the sum and difference of and ab??

a+b
2

2

Ans. Sum, a; difference, b.

8. What is the sum and difference of

(a+x)2 (a—x)2, and

2

?

Ans. Sum, a+x2; and difference, 2ax.

1

9. What is the sum and difference of and .?

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x+w

x-w

difference, -w

10. What is the product and quotient of

Ans. Product,

a+x

a-x

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and

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a- -X

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INVOLUTION AND EVOLUTION.

54. INVOLUTION is the raising of powers, and EvoLUTION the extraction of roots.

Involution is performed by the continued multiplication of the quantity into itself, till the number of factors amount to the number of units in the index of the power to which the quantity is to be involved; thus, the square of a is @Xa=a2; the 5th power of x is xxxxx=x5; the 5th power of (2a) is 2×2×2×2×2× aaaaa=32a5.

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Squares,
Cubes,

1

16

1 8 27

25 36 64 125 216 343 512 4th powers, 116 81 256 625 1296 2401 4096 6561 1 32 243 1024 3125 7776 16807 32768 59049

49

64

81

729

5th powers, 55. In raising simple algebraic quantities to any power, observe the following rules:-1st, Raise the numerical coefficient to the given power for the coefficient. 2d, Multiply the exponents of each of the letters by the power to which the quantity is to be raised. 3d, If the sign of the given quantity be plus, the signs of all the powers will be plus; but if the sign of the given quantity be minus, all the even powers will be plus, and the odd powers will be minus.

These rules are exemplified in the following Table, which the pupil should fill up for himself:

ROOTS AND POWERS OF SIMPLE ALGEBRAIC QUANTITIES.

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56. When it is required to raise a compound quantity to any power, it can always be effected by multiplying the quantity successively by itself; but there is another means of finding the powers of a binomial, commonly called the binomial theorem, first given in all its generality by Sir Isaac Newton, by which the power required can be written at once, without going through all the intermediate steps.

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